Sorry to keep up the questions, but from the belief revision article: (01)
To express the logic, a Tarskian consequence operator will be
used. Intuitively speaking, for any set A of sentences, Cn(A) is the set of
logical consequences of A. More formally, a consequence operation on a given
language is a function Cn from sets of sentences to sets of sentences. It
satisfies the following three conditions: (02)
Inclusion: A⊆Cn(A)
Monotony: If A⊆B, then Cn(A) ⊆ Cn(B)
Iteration: Cn(A) = Cn(Cn(A)) (03)
Cn is assumed to be supraclassical, i.e. if p can be derived
from A by classical truth-functional logic, then p ∈ Cn(A). A is a belief set
if and only if A = Cn(A). In what follows, K will denote a belief set. X ⊢ p
is an alternative notation for p ∈ Cn(X), and X ⊬ p for p ∉ Cn(X).
Cn(∅) is the set of tautologies. (04)
The expansion of K by a sentence p, i.e. the operation that
just adds p and removes nothing, is denoted K+p and defined as follows: K+p =
Cn(K∪{p}). (05)
It's not clear whether the "Tarskian Consequence Operator" Cn(A) is a sequence
or the entire closure of consequences of A. Is it defined to be Nth step in
recursing Cn(A) or is it the limiting consequence, or could it be both kinds? (06)
Does the innocent phrase "classical truth-functional logic" hide some
Hydeian(?) consequences? (07)
-Rich (08)
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5 - 5 7 1 2
-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Friday, January 02, 2015 9:54 AM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Cc: cg@xxxxxxxxxxxxx; Pavlos Peppas
Subject: Re: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf
Lindenbaum (09)
On 1/2/2015 11:13 AM, Rich Cooper wrote:
> That belief revision article is confusing when talking about
> “epistemic value” as a measure of “entrenchment”. (010)
Any document that uses the word 'epistemic' is likely to be confusing.
That comment is not a slur on the people who use the word 'epistemic'.
It's just an observation that nobody has ever been able to state a
precise definition of the words 'knowledge' and 'belief' that is
consistent with the way ordinary people use those words. (011)
However, the word 'entrenchment' as used in discussions of belief
or theory revision can be defined precisely. It just means that
you can specify a partial ordering of certain statements. (012)
The simplest ordering assumes two kinds of statements. For example,
they may be called laws and facts. The laws are more "entrenched"
than the facts. When you're revising a theory, you would preserve
the laws and revise the facts. (013)
Another pair of terms is T-Box (terminology) vs. A-Box (assertions).
The T-Box (AKA ontology) is more entrenched than the A-Box, and
any revisions should be made to the A-Box rather than the T-Box. (014)
According to Dunn's semantics for modal logic (which I recommend),
that two-way distinction can be used to define the modal operators.
Any statement implied by the laws (or by the T-Box) is defined to
be necessarily true. Any statement that is consistent with the
laws (or T-Box) is defined to be possible. (015)
In the following articles, I generalized Dunn's semantics to allow
a partial ordering of the laws by degree of entrenchment: (016)
http://www.jfsowa.com/pubs/laws.htm
Laws, facts, and contexts (017)
http://www.jfsowa.com/pubs/worlds.pdf
Worlds, models, and descriptions (018)
For example, the most entrenched laws would be "logically true"
(i.e., valid or true in all models). Next would be physically
true according to known laws of physics. Next would be the laws
of various authorities, such as the Bible, the US Constitution,
your mommy, etc. -- ordered according to your personal concerns. (019)
The meanings of the words 'necessary' or 'must' depend on which
laws are being considered in the current context. (020)
Summary: The reasons why I prefer Dunn's semantics and Lindenbaum
lattices are (a) they're consistent with the confusing publications
philosophers produce, (b) they're compatible with the implementations
used in AI, and (c) they're much, much easier to explain to students
and ordinary human beings. (021)
John (022)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J (023)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J (024)
|